Question

A merchant has 140 litres, 260 litres and 320 litres of three kinds of oil. He wants to sell the oil by filling the three kinds of oil separately in tins of equal volumes. The volume of such a tin is:

Answer 1

Hint: The greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted as\[gcd\left( {x,y} \right)\] .
The merchant wants to sell the oil by filling the three kinds of oil separately in tins of equal volumes, then the volume of such a tin is given by the largest number that divides each kind of oil volume which is nothing but the greatest common divisor of the three volumes.

Complete step-by-step answer:
It is given that the merchant has 140 litres, 260 litres and 320 litres of three kinds of oil.
Now if he wants to sell the oil by filling the three kinds of oil separately in tins of equal volumes then the merchant has to find the capacity of tin that divides the capacity of oil exactly.
So the volume of such a tin is given by the greatest common divisor of 140 litres, 260 litres and 320 litres.
To find the greatest common divisor we should find the prime factorization of all the litres,
Hence prime factorization of
\[140{\rm{ }} = 2 \times 2 \times 5 \times 7\]
\[260 = 2 \times 2 \times 5 \times 13\]
\[320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5\]
Hence the greatest common divisor of 140, 260,320 is
 \[gcd\left( {140,260,320} \right) = 2 \times 2 \times 5 = 20\]
Thus, the volume of required tin is 20 litres.
That is he can sell every oil using 20 litres tin.

Note: Here the greatest common divisor is found from the prime factorization of each number. On comparing the same numbers in the prime factorization we get the greatest common divisor.
That is, here 2, 2, 5 are common in every number hence the product is given as the greatest common divisor.